用频域方法提升神经微分方程对复杂动力系统的建模能力。
Frequency-Domain Neural ODEs for Modeling Non-Linear Dynamical Systems
- 将时间动态通过FFT转至频域建模，增强连续深度学习的表达力。
- 在4类动力系统上均优于GRU、LSTM及ANODE等模型，泛化性能更优。
- 适合需要高稳定性与泛化能力的动力系统建模任务。
标准连续深度模型（如神经微分方程，NODE）通过学习连续向量场而非离散时间步，在建模物理系统方面具有显著优势。然而，面对复杂动力系统时，标准NODE常难以处理高度非线性动态。本文提出频域神经微分方程（FNODE），利用快速傅里叶变换（FFT）将连续时间动态投影到频域。该架构在频域中操作，提升了对动力系统的泛化能力。实验在四个典型动力系统——洛特卡-沃尔泰拉模型、受迫杜芬振子、范德波尔振子和洛伦兹系统上，对比了FNODE与离散模型（如GRU、LSTM）及其他连续深度变体（如ANODE）。为严格评估泛化与鲁棒性，采用课程学习与集成学习，通过不同集成模型估计置信区间以评估收敛性。结果表明，FNODE在泛化性能上表现更优，且收敛稳定性显著提升。
原文摘要 · Abstract (English)
Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better generalization to the dynamical system. The architecture is empirically evaluated against discrete models, specifically Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs), and other continuous-depth variants, including Augmented Neural ODE (ANODE), across four distinct dynamical systems: the Lotka-Volterra model, the forced Duffing oscillator, the Van der Pol oscillator, and the Lorenz system. To rigorously assess generalization and robustness, curriculum and ensemble learning are used to evaluate the model's convergence by estimating confidence intervals across different ensemble models. The empirical results demonstrate that the FNODE architecture achieves better generalization while exhibiting remarkable convergence stability.
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